Extending Analytic Functions f:C->C , to f^:C^->C^; C^=Riemann Sphere


by WWGD
Tags: analytic, criemann, extending, fc>c, functions, sphere
WWGD
WWGD is offline
#1
Feb13-09, 12:51 AM
P: 391
Hi, everyone:
I am trying to see if it is true that if we are given f:C-->C analytic ; C complex plane,
then, to extend f to a function defined on C^ (Riemann Sphere) , i.e, to get:

f^: C^ -->C^


with f^|_C =f


i.e., the restriction of f^ to the complex plane agrees with f ,


If we need to define f(oo) =oo .


I think the answer is yes. Here's what I have:


We consider a 'hood ('hood:=neighborhood.) W of oo in C^ , which are complements
of compact 'hoods K in C , together with {oo}, i.e., W=C\K U {oo} , for K compact in C.

By Liouville's thm., |f|-->oo on balls B(0;r) , as r-->oo . And then by continuity,
it would seem that we need f(oo)=oo, since W= C-B(0;r) is a 'hood of oo.

Alternatively, if we had an analytic map f on C , f would go to oo on balls B(0;r)
as r->oo . Then if we used the stereo projection S: C-->C^ , and push f
forward by this projection, we would have Sof (composition)-->oo .


But this is still not rigorous-enough.

Any Ideas?.

Thanks.

P.S: If it bothers people to use regular ASCII, please let me know. I use ASCII
as a way to force myself to keep things clear . But I can change if neccessary.
Phys.Org News Partner Science news on Phys.org
Review: With Galaxy S5, Samsung proves less can be more
Making graphene in your kitchen
Study casts doubt on climate benefit of biofuels from corn residue

Register to reply

Related Discussions
Where are these functions analytic? Calculus & Beyond Homework 1
analytic functions Calculus & Beyond Homework 1
Mapping Analytic Functions Calculus & Beyond Homework 3
Geometry of Analytic Functions General Math 0
analytic functions proofs. Calculus & Beyond Homework 5